Mathematical Modeling

**Lectures:** Mondays and Wednesdays, 4:00-5:30 in
213 McLaughlin College.

**Office hours**
are posted here

**Texts:**

(1) *An Introduction to Mathematical Modeling*,
by Edward A. Bender

(2) *Mathematical Models: Mechanical Vibrations, Population Dynamics,
and Traffic Flow*, by Richard Haberman

Additional materials will be distributed in class.

**Evaluation:**

Homework : 65% Final exam (December): 35%

Other important information relevant to all York courses is posted here.

Slides from class of September 13, 2017

Excel spreadsheet for population models -- class of Sept 13

Additional notes on a random growth model -- for the class of Sept 13

One reference for SIR epidemic models is Section 2.1.2 in "Lecture Notes in Mathematical Epidemiology", edited by Fred Brauer, Pauline van den Driessche, and Jianhong Wu (Springer 2008); this is available as an online resource through the York University Library catalogue (see the chapter "Compartmental Models in Epidemiology")

There are many references for the basics of Markov chains,
online as well as in the library. One such electronic
text available through the York Library catalogue is
"Introduction to Probability Models" by Sheldon M. Ross (2007).
See Chapter 4, Markov Chains, especially sections 4.1, 4.2, and
4.4.

Examples of Markov chains -- class of
October 18

Definitions and properties
of Markov chains -- class of October 18 and 23

Markov chain "insurance" example;
review of confidence intervals -- class of October 23 and 25

Notes on the exponential distribution and
conctinuous-time random processes -- class of October 25

Additional notes on Monte Carlo:
Generating exponential and normal random variables
-- class of October 25 and 30

Diffusion: Notes on random walks
-- class of November 27

Problem Set 1 --- due Sept 20.

Comments on Problem Set 1

Problem Set 2 --- due Oct 2

Problem Set 3 --- due Oct 16

Problem Set 4 --- due Oct 30

Problem Set 5 --- due Nov 13 (and 20)

Problem Set 6 --- due Dec 4 (in class)

Here are the final exams from the last two times I taught this course: Fall 2010 and Fall 2011. Solutions will not be posted.

**
The final exam will be Friday December 15, 9:00 a.m. to 12:00 noon,
in Vari Hall 3006.** The course exam will be the same as the
comprehensive exam.

For the exam, you may bring a basic scientific calculator, with no graphing or programming or communications capabilities. You may also bring one sheet of notes with anything written on both sides. (You will need to hand in the sheet of notes at the end of the exam.)

Syllabus for the exam:

Main References:

E. A. Bender, "An Introduction to Mathematical Modelling", Dover, 1978.

A.C. Fowler, "Mathematical Models in the Applied Sciences", Cambridge
University Press, 1996.

R. Haberman, "Mathematical Models", SIAM, 1998.

Additional reference:

J.D. Murray, "Mathematical Biology", Springer, 1989 (or later edition).

TOPICS:

Principles of modelling (Bender)

Scaling, dimensional analysis (Bender, Fowler)

Graphical models (Bender)

Dynamical systems: discrete and continuous time; stability (Bender, Haberman)

Markov chains; simple continuous-time stochastic models (class materials,
Bender)

Monte Carlo methods (Bender)

Asymptotics (Fowler)

Perturbation of differential equations: boundary layers (Fowler, Murray)

Diffusion--deterministic and stochastic (class materials, Murray)

Basic population models: growth, interacting species
(Bender, Haberman, class materials)

Simple epidemic models (class notes, Murray)

Traffic models (Haberman)

**Office hours during the exam period:** (Subject to change:)

Thursday Dec 7: 1:00-2:00

Monday Dec 11: 11:00-12:00

Tuesday Dec 12: 1:30-2:30

Wednesday Dec 13: 2:00-3:00

Thursday Dec 14: 11:00-12:00

Last modified: December 1, 2017